A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

TL;DR

This paper introduces a formal Riemannian metric on conformal classes of Riemann surfaces, explores its properties, and studies a new geometric flow related to the inverse Gauss curvature, offering a novel approach to uniformization.

Contribution

It defines a new Riemannian structure on conformal classes and analyzes a related geometric flow, providing insights into the uniformization theorem via a novel inverse Gauss curvature flow.

Findings

The Riemannian metric has nonpositive curvature.

Liouville energy is geodesically convex.

The inverse Gauss curvature flow exists long-term and converges weakly to constant curvature.

Abstract

We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is geodesically convex. The geodesic equation for this metric corresponds to a degenerate elliptic fully nonlinear PDE, and we prove that any two points are connected by a $C^{1,1}$ geodesic. Using this we can define a length space structure on the given conformal class. We present a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics. This is all a special case of a more general construction on even dimensional manifolds related to the $\sigma_{\frac{n}{2}}$-Yamabe problem, which will appear in a forthcoming article.